To solve the quadratic equation \( 2x^2 + 4x + 1 = 0 \), we will use the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

In this equation, \( a = 2 \), \( b = 4 \), and \( c = 1 \).

### Step 1: Calculate the discriminant
\[
\Delta = b^2 - 4ac = 4^2 - 4(2)(1) = 16 - 8 = 8
\]

### Step 2: Apply the quadratic formula
\[
x = \frac{-4 \pm \sqrt{8}}{2(2)} = \frac{-4 \pm 2\sqrt{2}}{4}
\]

### Step 3: Simplify the expression
\[
x = \frac{-4}{4} \pm \frac{2\sqrt{2}}{4} = -1 \pm \frac{\sqrt{2}}{2}
\]

Thus, the two solutions are:
\[
x_1 = -1 + \frac{\sqrt{2}}{2}, \quad x_2 = -1 - \frac{\sqrt{2}}{2}
\]

Let me know if you need further details on any step.

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Here are 5 related questions:
1. How does the discriminant affect the nature of the roots?
2. Can you solve the equation \(3x^2 + 5x - 2 = 0\) using the quadratic formula?
3. What happens if the discriminant is negative in a quadratic equation?
4. How can you use completing the square to solve a quadratic equation?
5. Can this equation be factored, or is the quadratic formula necessary?

**Tip:** The discriminant \( \Delta = b^2 - 4ac \) determines whether the roots are real or complex. If \( \Delta > 0 \), the roots are real and distinct; if \( \Delta = 0 \), they are real and identical; if \( \Delta < 0 \), the roots are complex.

Discover how to solve the quadratic equation 2x² + 4x + 1 = 0 using the quadratic formula. This step-by-step guide explains the calculation of the discriminant and the application of the quadratic formula to find the solutions. Ideal for students in Grades 9-12.

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